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The analytic bootstrap equations of non-diagonal
two-dimensional CFT
Santiago Migliaccio, Sylvain Ribault
To cite this version:
Santiago Migliaccio, Sylvain Ribault. The analytic bootstrap equations of non-diagonal two-dimensional CFT. Journal of High Energy Physics, Springer, 2018, 2018, pp.169. �10.1007/JHEP05(2018)169�. �cea-01695664�
arXiv:1711.08916v1 [hep-th] 24 Nov 2017
The analytic bootstrap equations
of non-diagonal two-dimensional CFT
Santiago Migliaccio and Sylvain Ribault
Institut de physique théorique, Université Paris Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette
E-mail: santiago.migliaccio@ipht.fr, sylvain.ribault@ipht.fr
Abstract: Under the assumption that degenerate fields exist, diagonal CFTs such as Liouville theory can be solved analytically using the conformal bootstrap method. Here we generalize this approach to non-diagonal CFTs, i.e. CFTs whose primary fields have nonzero conformal spins. Assuming generic values of the central charge, we find that the non-diagonal sector of the spectrum must be parametrized by two integer numbers. We then derive and solve the equations that determine how three- and four-point structure constants depend on these numbers. In order to test these results, we numerically check crossing symmetry of a class of four-point functions in a non-rational limit of D-series minimal models. The simplest four-point functions in this class were previously argued to describe connectivities of clusters in the critical Potts model.
Contents
1 Introduction and summary 1
2 Spinful fields and their correlation functions 3
2.1 Global conformal symmetry 3
2.2 Diagonal and non-diagonal fields 4
3 Analytic conformal bootstrap 6
3.1 Operator product expansions and crossing symmetry 6
3.2 Ratios of four-point structure constants 8
3.3 Three-point structure constants 11
3.4 Relation with Liouville theory 14
4 Crossing-symmetric four-point functions 15
4.1 Non-rational limit of minimal models 16
4.2 Numerical tests of crossing symmetry 18
4.3 Case of the Ashkin–Teller model 21
5 Conclusion 22
A Universality of the geometric mean relation 23
A.1 Mathematical statement 23
A.2 Conformal field theory applications 24
1
Introduction and summary
The conformal bootstrap method is arguably the simplest way of exactly computing corre-lation functions in diagonal CFTs such as Liouville theory and minimal models. The idea is to constrain generic three-point structure constants by studying four-point functions that involve degenerate fields. This is possible not only if degenerate fields correspond to states in the model’s spectrum, but also under the weaker assumption that they merely exist, an assumption which holds in the case of Liouville theory. (See [1] for a brief review.)
In this article we will investigate whether the same method can be applied to the case of non-diagonal CFTs. There is ample motivation for investigating these theories: for example, the CFT that describes the Potts model at criticality is expected to be non-diagonal. However, we will not attempt to solve any specific model, but rather compute correlation functions that obey conformal bootstrap equations. In order to derive such equations, we will assume
2. that correlation functions are single-valued,
3. that the model depends analytically on the central charge.
As a first consequence of these assumptions, we will show that non-diagonal primary fields are parametrized by two integer numbers. (By non-diagonal fields we mean not only spinful fields, but also spinless fields that generate spinful fields when fused with degenerate fields.) Each degenerate field will be responsible for shifting the value of one of the integer numbers, and the resulting bootstrap equations will therefore determine the correlation functions of our non-diagonal fields.
Some steps in this direction were previously taken by Estienne and Ikhlef [2], who found the striking result that three-point structure constants of certain spinful fields were geometric means of Liouville three-point structure constants. (See also [3] for a similar relation in a more complicated CFT.) Schematically,
C(∆i, ¯∆i) =
q
CL(∆i)CL( ¯∆i) , (1.1)
where ∆i and ¯∆i are left- and right-moving conformal dimensions respectively, so that
a primary field is spinful if ∆i 6= ¯∆i and spinless if ∆i = ¯∆i. The structure constants
that we will compute do obey a geometric mean relation, where we will resolve the sign ambiguity and show that the square root is compatible with an analytic dependence on the central charge. In AppendixAwe will actually show that the geometric mean relation is a universal feature of non-diagonal CFTs under certain assumptions.
The bootstrap equations that determine the structure constants are derived from four-point functions that involve degenerate fields, but they do not imply that more general four-point functions are crossing-symmetric. In order to show that, we also need to determine the operator product expansions (OPEs) of the fields. And finding which fields appear in a given OPE is not necessarily easy. For instance, Liouville theory with a central charge less than one was shown to exist by the determination of its spectrum and OPEs [4] a long time after its structure constants were computed.
In order to guess plausible OPEs we will take a limit of the D-series Virasoro minimal models, a class of non-diagonal CFTs which have already been solved, but which exist only for discrete values of the central charge. Since however these values are dense in the half-line c ∈ (−∞, 1), taking limits of minimal model OPEs yields sensible OPEs for any c ∈ (−∞, 1), and actually by analyticity for any central charge in the half-plane
ℜc < 13 . (1.2)
By combining these OPEs and our analytic structure constants, we can compute four-point functions of two diagonal fields with arbitrary conformal dimensions, and two non-diagonal fields. We will numerically check that these four-point functions are crossing-symmetric. The D-series minimal models, along with their limits and analytic continuations, obey a rule of conservation of diagonality: a correlation function can be nonzero only if it involves an even number of non-diagonal fields. This implies that our four-point functions based on limits of D-series minimal models only involve three-point structure constants with 0 or 2 non-diagonal fields. In order to test our results with 1 or 3 non-diagonal
fields, we will focus on the Ashkin–Teller model, a c = 1 CFT where diagonality is not conserved. We will show that our analytic results agree with Al. Zamolodchikov’s [5] for this model.
2
Spinful fields and their correlation functions
2.1 Global conformal symmetry
Consider a field theory on the Riemman sphere, with local conformal symmetry. The symmetry algebra is made of two copies of the Virasoro algebra, with the same central charge c. For the moment, let us focus on global conformal transformations,
z → az + bcz + d , with a b c d
∈ SL2(C) . (2.1)
Under such transformations, a primary field V (z) with left and right conformal dimensions ∆ and ¯∆ behaves as
V (z) → (cz + d)−2∆(¯c¯z + ¯d)−2 ¯∆V az + b
cz + d
. (2.2)
In particular, a rotation by an angle θ corresponds to ei θ2 0 0 e−i θ2
, and gives
V (z) → e−iθ(∆− ¯∆)V eiθz . (2.3)
In this transformation, there appears the difference between the right and left dimensions, called the conformal spin and denoted as
S = ∆ − ¯∆ . (2.4)
In a CFT, correlation functions are invariant under conformal transformations. In partic-ular, the invariance of an n-point function of primary fields Vi(zi) with spins Si implies
* n Y i=1 Vi(zi) + = e−iθPnj=1Sj * n Y i=1 Vi(eiθzi) + . (2.5)
Assuming the correlation functions to be single-valued, the phase should be 1 for the rotation by θ = 2π, which implies
n
X
j=1
Sj ∈ Z . (2.6)
Let us consider the cases of two-point functions. Global conformal symmetry implies that a two-point function vanishes unless the two fields have the same left and right conformal dimensions. We further assume that there is an orthogonal basis of primary fields, so that two-point functions of elements of this basis take the form
D V1(z1)V2(z2) E = B1 δ12 z2∆112 z¯122 ¯∆1 , (2.7)
where B1 = B(V1) is a z-independent factor called the two-point structure constant. In
particular, the two-point function can be nonzero only if S1 = S2. Combined with the
constraint on the total spin eq. (2.6), this implies that the spins S1, S2 obey
S ∈ 12Z. (2.8)
All our fields will obey this constraint, and will be called bosons if S ∈ Z, and fermions if S ∈ Z
2.
In the case of three-point functions, global conformal symmetry implies
D V1(z1)V2(z2)V3(z3) E = C123 × z12∆3−∆2−∆1z¯ ¯ ∆3− ¯∆2− ¯∆1 12 z23∆1−∆2−∆3z¯ ¯ ∆1− ¯∆2− ¯∆3 23 z31∆2−∆3−∆1z¯ ¯ ∆2− ¯∆3− ¯∆1 31 , (2.9)
where C123 = C(V1, V2, V3) is called the three-point structure constant. Since fermions
anticommute, under a permutation σ the three-point function should behave as D Vσ(1)Vσ(2)Vσ(3) E = η123(σ) D V1V2V3 E , (2.10) where η123(σ) = (
−1 if σ exchanges two fermions,
1 else. (2.11)
In order to compensate for the behaviour of the z-dependent factor of the three-point function (2.9), the structure constant C123 should therefore satisfy
Cσ(1)σ(2)σ(3)
C123
= η123(σ) sgn(σ)S1+S2+S3 . (2.12)
2.2 Diagonal and non-diagonal fields
In order to determine the three-point structure constants, we will use constraints coming from four-point functions that involve degenerate fields. We assume that these degenerate fields exist, but they may well be unphysical, i.e. the corresponding states may not be part of the spectrum.
Before writing fusion rules involving degenerate fields, let us introduce notations that make them simpler. We have the alternative notation β for the central charge c,
c = 1 − 6
β − β1 2
, (2.13)
and we introduce the momentum P instead of the conformal dimension ∆,
∆ = c − 1 24 + P
2. (2.14)
In terms of momentums, the spin is
Writing VP, ¯P a primary field with left and right momentums P and ¯P , the fusion product
of a left and right degenerate field Vh2,1i with such fields takes the form
Vh2,1i× VP, ¯P ⊆
X
ǫ=±, ¯ǫ=±
V
P +ǫβ2 , ¯P +¯ǫβ2 . (2.16)
This fusion rule follows from the existence of vanishing descendents of Vh2,1i. Let us
furthermore assume that the fields satisfy the half-integer spin condition (2.8). Then the difference of the spins of VP, ¯P and V
P +ǫβ2, ¯P +ǫβ¯2 must be half-integer, which implies
P − σ ¯P ∈ 1
2βZ, (2.17)
with σ = ǫ¯ǫ ∈ {+, −}. If this holds for both values of σ, then it follows that S =Y
±
(P ± ¯P ) ∈ 4β12Z. (2.18)
Assuming that S 6= 0 and that the central charge is generic, so that β2 ∈ Q, this is/
incompatible with the half-integer spin condition (2.8). Therefore, eq. (2.17) can be satisfied for only one value of σ, and the terms in the fusion product Vh2,1i× VP, ¯P with the
other value of σ must, in fact, be absent. Similarly, the fusion product Vh1,2i× VP, ¯P can
have only two terms, and these terms are determined by the value of the sign ˜σ ∈ {+, −} such that
P − ˜σ ¯P ∈ β2Z. (2.19)
We consider σ and ˜σ as properties of fields, which control their fusion products with degenerate fields: Vh2,1i× VP, ¯σ,˜Pσ = X ǫ=± Vσ,˜σ P +ǫβ2, ¯P +σǫβ2 , Vh1,2i× V σ,˜σ P, ¯P = X ǫ=± Vσ,˜σ P −2βǫ , ¯P −σǫ2β˜ . (2.20)
The fields on the right-hand sides have the same values of σ, ˜σ as the field VP, ¯σ,˜Pσ, because their momentums satisfy the same relations (2.17),(2.19). Let us now solve these relations and determine the momentums. The solution strongly depends on σ˜σ, and we will call fields diagonal or non-diagonal depending on this sign. Once this sign is fixed, the choice of σ makes little difference, and we will set σ = + without loss of generality.
• Diagonal fields ˜σ = σ
Still assuming β2 ∈ Q, diagonal fields must have P − σ ¯/ P ∈ β 2Z∩
1
2βZ = {0}, and
therefore ∆ = ¯∆. Introducing the notation
VPD = VP,P+,+ , (2.21)
the fusion products with degenerate fields read
Vh2,1i× VPD = X ǫ=± VP +D ǫβ 2 , Vh1,2i× V D P = X ǫ=± VP −D ǫ 2β . (2.22)
• Non-diagonal fields ˜σ = −σ
Keeping in mind the half-integer spin condition (2.8), the momentums of non-diagonal fields are of the type
P = P(r,s),
¯
P = σP(r,−s), with r, s, rs ∈
1
2Z, (2.23)
where we introduced the notation
P(r,s) = 1 2 rβ − s β . (2.24)
The spin of a non-diagonal field is S = −rs. Introducing the notation
V(r,s)N = VP(r,s),P(r,−s)+,− , (2.25) the fusion products with degenerate fields read
Vh2,1i× V(r,s)N = X ǫ=± V(r+ǫ,s)N , Vh1,2i× V(r,s)N = X ǫ=± V(r,s+ǫ)N . (2.26)
We emphasize that it is not enough to know the momentums of a field to identify it as diagonal or non-diagonal. The non-diagonal field VN
(0,s) has spin zero, and the same left
and right momentums as the diagonal field VD
P(0,s). These two fields are distinguished by
their fusion products with degenerate fields, equivalently by their values of σ and ˜σ. In particular, the fusion product
Vh2,1i× V(0,s)N =
X
ǫ=±
V(ǫ,s)N , (2.27)
can produce spinful fields.
3
Analytic conformal bootstrap
In this Section we will derive and solve conformal bootstrap equations for correlation functions of our diagonal and non-diagonal fields. These equations will follow from the assumption that the correlation functions Vh2,1iV1V2V3 and Vh1,2iV1V2V3 exist, for
ar-bitrary diagonal or non-diagonal fields V1, V2, V3.
3.1 Operator product expansions and crossing symmetry
We will write operator product expansions in a schematic notation that omits the depen-dence on the coordinates zi, and also omits the contributions of descendent fields:
V1V2 =
X
V3∈S
C123 V3. (3.1)
Here S is a set of primary fields that we call the spectrum of the OPE. The spectrum of an OPE is a subset of the spectrum of the theory. Inserting the OPE into the three-point
function (2.9), we find that the OPE coefficient C3
12 is related to the two- and three-point
structure constants B3 (2.7) and C123 by
C123 = B3C123 . (3.2)
Let us write the degenerate OPEs that correspond to the fusion rules (2.20) as
Vh2,1iV = X ǫ=± Cǫ(V )Vǫ , Vh1,2iV = X ǫ=± ˜ Cǫ(V )V˜ǫ, (3.3)
where we introduced the notations
V = VP, ¯σ,˜Pσ ⇒ Vǫ = Vσ,˜σ
P +ǫβ2, ¯P +σǫβ2 , V ˜
ǫ = Vσ,˜σ
P −2βǫ , ¯P −2βσǫ˜ , (3.4)
and the degenerate OPE coefficients Cǫ(V ), ˜Cǫ(V ). The existence of OPEs implies that
four-point functions can be decomposed into combinations of conformal blocks. In par-ticular, an s-channel decomposition is obtained by inserting the OPE of the fields V1 and
V2, and reads D V1V2V3V4 E = X Vs∈Ss Ds|1234F∆s(s)(∆i|zi)F∆s(s)¯ ( ¯∆i|¯zi) . (3.5)
This is a combination of four-point conformal blocks F∆s(s)(∆i|zi), with the four-point
s-channel structure constants 2 s 3 1 4 Ds|1234= C12s Cs34 = C12sCs34 Bs . (3.6)
Alternatively, we could insert the OPE of the fields V1 and V4, and obtain the t-channel
decomposition of the same four-point function. Both decompositions should agree, and we obtain the crossing symmetry equation
X Vs∈Ss Ds|1234 2 s 3 1 4 = X Vt∈St Dt|4123 1 4 t 3 2 (3.7)
Let us now consider a four-point function that involves at least one degenerate field Vh2,1i.
Since the OPEs of this field have only two terms, its four-point functions involve only two terms in each channel,
D Vh2,1iV1V2V3 E = X ǫ1=± d(s)ǫ1 Fǫ1(s)F¯σ1ǫ1(s) = X ǫ3=± d(t)ǫ3Fǫ3(t)F¯σ3(t)ǫ3 , (3.8)
where Fǫ(s), Fǫ(t) are degenerate four-point conformal blocks, and the four-point structure
constants are of the type
1 1ǫ 2 h2, 1i 3 d(s)ǫ = Cǫ(V1) C(V1ǫ, V2, V3) . (3.9)
Let us introduce the ratio ρ = ρ(V1|V2, V3) = d(s)+ d(s)−
, which is important because we will be able to compute it in Section 3.2. Its expression in terms of structure constants is
ρ(V1|V2, V3) =
C+(V1)C(V1+, V2, V3)
C−(V1)C(V1−, V2, V3)
. (3.10)
In the particular case of the four-point function Vh2,1iV1Vh2,1iV1 , the four-point structure
constants are of the type
1 1ǫ 1 h2, 1i h2, 1i d(s)ǫ = Cǫ(V1) B(V1ǫ) Cǫ(V1) , (3.11)
and the corresponding ratio is
ρ(V1) =
C+(V1)2B(V1+)
C−(V1)2B(V1−)
. (3.12)
It follows that the dependence of the four-point structure constants Ds|1234 = D(Vs) (3.6)
on the field Vs obeys
D(V+ s ) D(V− s ) = ρ(Vs|V1, V2)ρ(Vs|V3, V4) ρ(Vs) , D(V ˜ + s ) D(V−˜ s ) = ρ(V˜ s|V1, V2)˜ρ(Vs|V3, V4) ˜ ρ(Vs) , (3.13)
where the second equation is obtained by replacing Vh2,1i with Vh1,2i in our analysis. Next
we will determine the ratios ρ and ˜ρ of degenerate four-point structure constants.
3.2 Ratios of four-point structure constants
The s- and t-channel degenerate four-point conformal blocks Fǫ(s) and Fǫ(t) that appear in
the decomposition (3.8) of Vh2,1iV1V2V3 are two bases of solutions of the same Belavin–
Polyakov–Zamolodchikov equation [6]. They are related by a change of basis of the type
Fǫ1(s)=
X
ǫ3
whose coefficients are the fusing matrix elements Fǫ1,ǫ3 = Γ(1 + 2βǫ1P1)Γ(−2βǫ3P3) Q ±Γ(12 + βǫ1P1± βP2− βǫ3P3)) . (3.15)
The right-moving blocks ¯Fǫ(s) and ¯Fǫ(t) obey a similar relation, with a fusing matrix ¯F
whose elements are obtained from those of F by Pi → ¯Pi. As a consequence, we obtain
four equations relating the structure constants d(s)± , d (t)
± of both channels, two for each
value of ǫ3 = ±: X ǫ1 d(s)ǫ1 Fǫ1,ǫ3F¯σ1ǫ1,−σ3ǫ3 = 0 , (3.16) X ǫ1 d(s)ǫ1 Fǫ1,ǫ3F¯σ1ǫ1,σ3ǫ3 = d(t)ǫ3 . (3.17)
The four-point function vanishes unless equations (3.16) admit a non-zero solution for d(s)ǫ , which happens only if
F++F−− F+−F−+ = ¯ F++F¯−− ¯ F+−F¯−+ σ1σ3 . (3.18)
Explicitly, this condition reads
Y ± cos πβ(P1± P2− P3) cos πβ(P1± P2+ P3) =Y ± cos πβ(σ1P¯1± ¯P2− σ3P¯3) cos πβ(σ1P¯1± ¯P2+ σ3P¯3) . (3.19)
Assuming that our three fields are diagonal or non-diagonal as discussed in Section 2.2, the numbers si = β(σiP¯i− Pi) must be half-integer. (In particular, we are now defining
s = 0 for a diagonal field.) Then our condition reduces to
1 − (−1)2P3i=1si
sin(2πβP1) cos(2πβP2) sin(2πβP3) = 0 . (3.20)
Assuming we are in the generic situation where the trigonometric factors do not vanish, this implies
3
X
i=1
si ∈ Z . (3.21)
Now, since fusion with Vh2,1i (2.20) leaves the number s unchanged, this condition holds
not only for our four-point functionVh2,1iV1V2V3 , but also for the three-point functions
that result from the fusion Vh2,1i × V1. It follows that any nonzero three-point function
must obey this condition. The same analysis with the correlation function Vh1,2iV1V2V3 ,
and with the convention r = 0 for a diagonal field, leads to the analogous condition
3
X
i=1
ri ∈ Z . (3.22)
• For any three-point function of the type VDVDVN , the non-diagonal field must
have integer indices r, s ∈ Z.
• Any three-point function with an odd number of fermionic fields vanishes, because fermionic fields obey ri + si ∈ Z + 12 while P3i=1(ri + si) ∈ Z. So our conditions
imply the single-valuedness condition (2.6) for three-point functions.
Returning to the four-point function Vh2,1iV1V2V3 we see that, if conditions (3.18) are
obeyed, the ratios of the four-point structure constants are given in terms of fusing matrix elements, ρ = d (s) + d(s)− = − F−,ǫ3F¯−σ1,−σ3ǫ3 F+,ǫ3F¯σ1,−σ3ǫ3 , ∀ǫ3 ∈ {+, −} . (3.23)
Inserting explicit expressions (3.15) for the fusing matrix elements, we obtain
ρ(V1|V2, V3) = −Γ(−2βP1 ) Γ(2βP1) Γ(−2βσ1P¯1) Γ(2βσ1P¯1) ×Y ± Γ(12 + βP1± βP2+ βǫ3P3) Γ(12 − βP1± βP2+ βǫ3P3) Y ± Γ(12 + βσ1P¯1± β ¯P2− βǫ3σ3P¯3) Γ(12 − βσ1P¯1 ± β ¯P2− βǫ3σ3P¯3) , (3.24)
where we have restored the explicit dependence of ρ = ρ(V1|V2, V3) on the fields.
Remem-bering that si = β(σiP¯i− Pi) ∈ 12Z obey eq. (3.21), this can be rewritten in manifestly
ǫ3, σ3-independent way: ρ(V1|V2, V3) = −(−1)2s2Γ(−2βP1 ) Γ(2βP1) Γ(−2βσ1P¯1) Γ(2βσ1P¯1) Y ±± Γ(12 + βP1± βP2± βP3) Γ(12 − βσ1P¯1± β ¯P2± β ¯P3) . (3.25)
Let us check that this behaves as expected when we permute the fields V2 and V3. Using
eq. (3.21), we find
ρ(V1|V2, V3) = (−1)2s1ρ(V1|V3, V2) . (3.26)
This is actually what we expect from the expression of ρ(V1|V2, V3) in terms of
struc-ture constants (3.10), given their behaviour (2.12) under permutations, and the relation (−1)2s1 = (−1)S(V1+)−S(V1−).
The analogous expression for the ratio ˜ρ = d˜
(s) + ˜
d(s)− of s-channel structure constants of the
four-point function Vh1,2iV1V2V3 is obtained by the substitutions s → r, σ → ˜σ, β →
−β−1: ˜ ρ(V1|V2, V3) = −(−1)2r2 Γ(2β−1P 1) Γ(−2β−1P 1) Γ(2β−1σ˜ 1P¯1) Γ(−2β−1σ˜ 1P¯1) ×Y ±± Γ(1 2 − β −1P 1± β−1P2± β−1P3) Γ(1 2 + β−1σ˜1P¯1 ± β−1P¯2± β−1P¯3) . (3.27)
The particular case of a four-point function Vh2,1iV Vh2,1iV with two degenerate fields corresponds to (P1, ¯P1) = (P3, ¯P3) = (P, ¯P ) , (3.28) P2 = ¯P2 = β − 2β1 , (3.29) and we find ρ(V ) = −Γ(−2βP )Γ(−2βσ ¯P ) Γ(2βP )Γ(2βσ ¯P ) Γ(β2+ 2βP )Γ(1 − β2+ 2βP ) Γ(β2− 2βσ ¯P )Γ(1 − β2− 2βσ ¯P ) . (3.30)
Similarly, the ratio of structure constants of the four-point function Vh1,2iV Vh1,2iV is
˜ ρ(V ) = − Γ(2β −1P )Γ(2β−1σ ¯˜P ) Γ(−2β−1P )Γ(−2β−1σ ¯˜P ) Γ(β−2− 2β−1P )Γ(1 − β−2− 2β−1P ) Γ(β−2+ 2β−1˜σ ¯P )Γ(1 − β−2+ 2β−1σ ¯˜P ). (3.31)
These ratios slightly simplify in the case of diagonal fields i.e. if P = σ ¯P = ˜σ ¯P ,
ρ(VPD) = − Γ2(−2βP ) Γ2(2βP ) γ(β2+ 2βP ) γ(β2− 2βP ), ρ(V˜ D P ) = − Γ2(2β−1P ) Γ2(−2β−1P ) γ(β−2− 2β−1P ) γ(β−2+ 2β−1P ), (3.32) where we introduced γ(x) = Γ(1−x)Γ(x) .
The determination of the ratios ρ, ˜ρ lead to shift equations (3.13) for the four-point structure constants that appear in generic four-point functions hV1V2V3V4i. For some
spectrums, these shift equations are enough for determining the structure constants up to a Vs-independent factor, and therefore enough for checking crossing symmetry. We will
see examples of this in Section 4. For the moment, let us determine three-point structure constants.
3.3 Three-point structure constants
Before studying how our conformal bootstrap equations constrain three-point structure constants, let us warn that in principle such structure constants cannot be fully deter-mined. This is because our fields, and therefore also their correlation functions, are only defined up to a z-independent renormalization,
Vi(z) → λiVi(z) . (3.33)
We could fix this ambiguity by imposing additional constraints on correlation functions, as is done in minimal models by imposing B(V ) = 1, or in Liouville theory by imposing C+(V ) = ˜C+(V ) = 1 [6]. However, in the case of non-diagonal fields, imposing either
constraint would lead to correlation functions not being analytic in β. Rather than looking for an appropriate constraint, we will refrain from fixing the normalization. Therefore, we will only be able to determine normalization-independent quantities. Let us write three-point structure constant as
C123 = 3 Y i=1 Yi ! C123′ , (3.34)
where Yiis a normalization factor that behaves as Yi → λiYi. Since the two-point structure
constant (2.7) behaves as Bi → λ2iBi, the normalization-independent quantities that we
can hope to determine are
C123′ and Yi2B−1i . (3.35)
Rewriting the four-point s-channel structure constant as
Ds|1234= 4 Y i=1 Yi ! C12s′ Cs34′ Y2 s Bs , (3.36)
shows that the crossing symmetry equation (3.7) only involves these quantities.
The equations that constrain the structure constants are written in terms of the ratios ρ (3.10) and ˜ρ. If we define the normalization-independent ratio
ρ′(V1|V2, V3) = C′(V+ 1 , V2, V3) C′(V− 1 , V2, V3) , (3.37)
then the combination ρ(V1|V2, V3) ρ′(V 1|V2, V3) = C+(V1)Y (V + 1 ) C−(V1)Y (V1−) , (3.38)
should only depend on the field V1. So let us look for a function C′(V1, V2, V3) such that
the corresponding combinations ρρ′ and ˜ ρ ˜
ρ′ only depend on V1. The problem simplifies
significantly if at least one of the fields is diagonal, say V1 = VP1D, and we will restrict to
this case. We propose the ansatz
C′ VP1D, V2, V3 = f2,3(P1) Q ±,±Γβ( β 2 + 1 2β + P1± P2± P3)Γβ( β 2 + 1 2β − P1± ¯P2± ¯P3) , (3.39) where Γβ(x) is a double Gamma function with periods β and β−1, which is invariant under
β → β−1 and obeys Γβ(x + β) = √ 2πβ βx−12 Γ(βx)Γβ(x) . (3.40)
The factor f2,3(P1) ∈ {−1, +1} is included in order to account for the sign factors
appear-ing in eqs. (3.25) and (3.27). It is determined by the equations f2,3(P1+β2) f2,3(P1− β2) = (−1) 2s2 , f2,3(P1+ 1 2β) f2,3(P1−2β1 ) = (−1) 2r2. (3.41)
If r2, s2 ∈ Z, in particular if either V2 or V3 is diagonal, we have f2,3(P1) = 1. With our
ansatz, the ratios are
ρ′(V1D|V2, V3) = (−1)2s2β−8βP1 Y ±,± Γ 12 + βP1± βP2± βP3 Γ 1 2 − βP1± β ¯P2± β ¯P3 , (3.42) ˜ ρ′(V1D|V2, V3) = (−1)2r2β− 8 βP1 Y ±,± Γ 1 2 − β −1P 1± β−1P¯2± β−1P¯3 Γ 12 + β−1P 1± β−1P2± β−1P3 , (3.43)
which indeed match the expressions (3.25) and (3.27) for ρ(V1|V2, V3) and ˜ρ(V1|V2, V3), up
to factors that depend on V1 only.
Using the combination ρρ′ (3.38), together with the ratio ρ(V1) (3.12), we can write a
shift equation for the normalization-independent quantity Y2B−1,
Y2B−1(V+ 1 ) Y2B−1(V− 1 ) = ρ 2(V 1|V2, V3) ρ′2(V 1|V2, V3)ρ(V1) . (3.44)
In the case of a diagonal field, this equation is explicitly
(Y2B−1)VD,+ 1 (Y2B−1)VD,− 1 =−β 16βP1Γ2(−2βP1) Γ2(2βP 1) γ(β2− 2βP 1) γ(β2+ 2βP 1) , (3.45)
and the dual shift equation is
(Y2B−1)VD, ˜+ 1 (Y2B−1)VD, ˜− 1 =−β 16 βP1 Γ 2(2β−1P 1) Γ2(−2β−1P 1) γ(β−2+ 2β−1P 1) γ(β−2− 2β−1P 1) . (3.46)
A solution can be written in terms of the function Υβ(x) = Γβ(x)Γβ(β+β1 −1−x) as
(Y2B−1)(VPD) = Q 1
±Υβ(β ± 2P )
. (3.47)
Let us now determine the factor Y2B−1 for a field V
2 that is not necessarily diagonal.
We set σ2 = −˜σ2 = 1, but the result does not depend on these choices. We compute the
ratios ρ(V2|V3, V1D) and ˜ρ(V2|V3, V1D) by renaming the momentums in (3.25) and (3.27).
Making use of the permutation properties of the three-point structure constant (2.12) we rewrite ρ′(V2|V3, V1D) = C′(VD 1 , V2+, V3) C′(VD 1 , V2−, V3) , ρ˜′(V2|V3, V1D) = C′(VD 1 , V ˜ + 2 , V3) C′(VD 1 , V ˜ − 2 , V3) . (3.48)
Using the ansatz (3.39), we compute these ratios. Inserting them into eq. (3.44), we obtain (Y2B−1) V2+ (Y2B−1) V− 2 =−β 8β(P2+ ¯P2)Γ(−2βP2)Γ(−2β ¯P2) Γ(2βP2)Γ(2β ¯P2) × Γ(β 2− 2β ¯P 2)Γ(1 − β2− 2β ¯P2) Γ(β2+ 2βP 2)Γ(1 − β2+ 2βP2) , (3.49) (Y2B−1)V+˜ 2 (Y2B−1)V−˜ 2 = −β 8β−1(P2− ¯P2)Γ(2β−1P2)Γ(−2β−1P¯2) Γ(−2β−1P 2)Γ(2β−1P¯2) × Γ(β −2− 2β−1P¯ 2)Γ(1 − β−2− 2β−1P¯2) Γ(β−2− 2β−1P 2)Γ(1 − β−2− 2β−1P2) . (3.50)
These equations have the solution
(Y2B−1)VP, ¯σ,˜Pσ= (−1)P2− ¯P2Y
±
Γβ(β ± 2P )Γβ(β−1± 2 ¯P ) , (3.51)
which reduces to the diagonal case expression (3.47) if P = ± ¯P .
So we have determined the normalization-independent quantities C′ and Y2B−1, with
the exception of three-point structure constants that involve three non-diagonal fields. The formulas are slightly more complicated in that case, and we leave them for future work. Notice however that the shift equations (3.13) are enough for practical purposes, as we will demonstrate in an example in Section 4.3.
3.4 Relation with Liouville theory
Let us investigate whether our structure constants obey a relation of the type (1.1) with structure constants of Liouville theory. Let us start with the case of a three-point structure constant of three diagonal fields. In this case, the normalization-independent factor (3.39) reduces to CL′(P1, P2, P3) = Y ±± Υβ β 2 + 1 2β + P1± P2± P3 . (3.52)
For c ≤ 1 i.e. β ∈ R, this coincides with the analogous quantity in Liouville theory, as reviewed for example in [6]. For other values of c though, we have to use a different solution of the shift equations in order to recover Liouville theory results. Similarly, our solution for (Y2B−1)(VD
P ) (3.47) agrees with Liouville theory only if c ≤ 1, and has to be
replaced with the solution (Y2B−1)
L(P ) = c /∈(−∞,1)
Q
±Υiβ(±2iP ) otherwise. We will not
elaborate on this subtlety, because it affects only diagonal fields with their continuous values of the momentum P . Liouville theory structure constants that involve at least one discrete momentum of the type P(r,s) are determined by shift equations, modulo finitely
many initial conditions. This implies in particular that they are analytic functions of β, with no singularity at β ∈ R.
So, if a least one momentum is discrete, we can use the β ∈ R Liouville theory formula (3.52) for complex values of β as well. Then it is straightforward to check that our normalization-independent factor (3.39) obeys
C′2(V1D|V2, V3) = CL′(P1, P2, P3)CL′(P1, ¯P2, ¯P3) . (3.53)
Let us check the analogous relation for the quantity Y2B−1 (3.51). We will need the
identity (Y2B−1)VN (r,s) (Y2B−1)VN (r,−s) = Y ± Sβ(β ± 2P(r,s)) Sβ(β ± 2P(r,s)) = 1 , (3.54)
where we have introduced the special function Sβ(x) =
Γβ(x)
Γβ(β+β−1−x), which is invariant
under β → β−1 and obeys
Sβ(x + β)
Sβ(x)
We therefore have (Y2B−1) VN (r,s) 2 = (Y2B−1) V(r,s)N (Y2B−1) VN (r,−s) , (3.56)
and using ¯P(r,s)= P(r,−s) we easily obtain
(Y2B−1) VN (r,s) 2 = (Y2B−1)VPD(r,s)(Y2B−1)VP¯D(r,s) . (3.57)
Together with eq. (3.53), this shows that the geometric mean relation (1.1) holds at the level of normalization-independent quantities. We have actually written the square of this relation, in order to avoid having sign ambiguities. These sign ambiguities make the geometric mean relation more suggestive than practically useful. Still, squares of three-point structure constants do appear in four-three-point structure constants of the type Ds|1212,
and we find
Ds|1212= Y2(V1)Y2(V2)(Y2B−1)(Vs)CL′(P1, P2, Ps)C ′
L( ¯P1, ¯P2, ¯Ps) . (3.58)
This shows that the crossing symmetry equations for four-point functions of the type hV1V2V1V2i can be written in terms of Liouville three-point structure constants.
Our structure constants, and the shift equations that they solve, are ultimately derived from the fusing matrix. Therefore, the relation with Liouville theory should be expressible in terms of this matrix. Let us indeed write the square of the ratio (3.23) of fusing matrix elements, as the product of the two equivalent expressions for this ratio:
ρ2 = Y ǫ3=± F+,ǫ3F¯σ1,−σ3ǫ3 F−,ǫ3F¯−σ1,−σ3ǫ3 = F++F+− F−+F−− ¯ F++F¯+− ¯ F−+F¯−− σ1 . (3.59)
This relation can be rewritten as
ρ2(V1|V2, V3) = ρL(P1|P2, P3)ρσ1L ( ¯P1| ¯P2, ¯P3) , (3.60)
where ρL = F++F+−F−+F−− is the expression for ρ when all three fields are diagonal. This
relation implies the geometric mean relation for shift equations and therefore for structure constants. See Appendix A for a discussion of how general the geometric mean relation might be.
4
Crossing-symmetric four-point functions
In order to compute a four-point function, we need not only structure constants, but also a spectrum. But the spectrum of an OPE of two non-degenerate fields is a priori not easy to determine. In order to find plausible guesses for the spectrums of some OPEs, we will start with known OPEs in minimal models, and send the central charge to non-rational values.
4.1 Non-rational limit of minimal models
Let us first illustrate our approach in the case of diagonal models. For any coprime integers p, q ≥ 2, there exists a diagonal (A-series) minimal model with the parameter β2 = p
q. Its
spectrum is built from degenerate representations Rhr,si of the Virasoro algebra,
SA-series p,q = 1 2 q−1 M r=1 p−1 M s=1 Rhr,si 2 , (4.1)
where by |R|2 = R ⊗ ¯R we mean a representation R of the left-moving Virasoro algebra,
tensored with the same representation of the right-moving Virasoro algebra. Let us con-sider the limit of this spectrum as p, q → ∞ such that pq → β
2
0, for a given real value of β0.
Assuming β2
0 ∈ Q, the momentums P/ hr,si(2.24) of the states become dense in the real line.
Moreover, for a generic momentum P0 ∈ R, we have Phr,si → P0 ⇒ r, s → ∞. It follows
that the levels of the null vectors of Rhr,si go to infinity, and that lim
Phr,si→P0R
hr,si = VP0,
where VP0 is the Verma module of momentum P0. To summarize, the minimal models’
spectrums tend to a diagonal, continuous spectrum made of Verma modules,
lim p q→β 2 0 Sp,qA-series = Z R+dP |V P|2 . (4.2)
We thus recover the spectrum of Liouville theory. Since the three-point structure constants of Liouville theory and minimal models are analytic in β, P and obey the same shift equations, we conjecture that the correlation functions of Liouville theory with β ∈ R i.e. c ≤ 1 are limits of correlation functions of diagonal minimal models. Notice however that the spectrum of Liouville theory with c ≤ 1 was found much later than its structure constants [4]. Taking limits of minimal models is a shortcut that would have led to the correct spectrum, and that we will now use in the case of non-diagonal models.
For any coprime integers p, q such that q ≥ 6 is even and p ≥ 3 is odd, there exists a non-diagonal (D-series) minimal model with the parameter β2 = p
q [7]. The structure
of the spectrum depends on whether q ≡ 2 mod 4: for simplicity we assume that this condition is fulfilled. This assumption does not prevent the values of β from being dense in the real line, and does not affect our results. The spectrum is
Sp,qD-series q≡2 mod 4= 1 4 q−1 M r=12 p−1 M s=1 Rhr,si⊕ Rhq−r,si 2 , (4.3)
where r runs over odd integers. In this spectrum, half of the representations are diagonal, and the other half are non-diagonal, explicitly
SD-series p,q = q≡2 mod 4 1 2 q−1 M r=12 p−1 M s=1 Rhr,si 2 ⊕ 1 2 q−1 M r=12 p−1 M s=1 Rhr,si⊗ ¯Rhq−r,si . (4.4)
In particular, the representation Rhq
2,si
2
appears with the multiplicity 2, with one copy in the diagonal sector, and the other copy in the non-diagonal sector. The fusion rules of
the D-series minimal models have the remarkable property that diagonality is conserved [8],
D × D = D , D × N = N , N × N = D . (4.5) This property, plus the fusion products of degenerate representations, are enough for determining the spectrums of the OPEs in D-series minimal models.
In the non-diagonal sector of the spectrum SD-series
p,q , spins take the values
S = ∆(q−r,s)− ∆(r,s) = r − q 2 s − p 2 . (4.6)
These spins are integer, and must therefore remain constant when we take a limit p, q → ∞ such that pq → β2
0. This suggests that we take both factors r −q2 and s − p
2 to be constant,
and assume that r, s take p, q-dependent values, r = q 2 + r0 , s = p2 + s0 , with r0 ∈ 2Z , s0 ∈ Z +12 . (4.7)
Then ∆(r,s) = ∆(r0,s0) and ∆(q−r,s) = ∆(r0,−s0), so not only the spins but also the left
and right dimensions of our states remain constant in our limit. On the other hand, the diagonal sector behaves just like the spectrum of a diagonal minimal model in this limit. To summarize, lim p q→β20 SD-series p,q = 1 2 Z R dP |VP|2⊕ S2Z,Z+1 2 , (4.8)
where we use the notation SX,Y =Lr∈X
L
s∈Y VP(r,s) ⊗ ¯VP(r,−s).
Let us investigate how correlation functions behave in our limit. We write a four-point function as a sum over some spectrum in some channel. The sum is finite in minimal models, and becomes infinite in our limit. The convergence of the infinite sum depends on the behaviour of its terms as r, s → ∞. These terms are known to behave as decreasing exponentials in the total conformal dimension ∆ + ¯∆ as ∆ + ¯∆ → ∞. Now, if the limiting spectrum is non-diagonal and of the type SX,Y, then the total dimension of
a state is ∆(r,s)+ ∆(r,−s) = c − 1 12 + 1 2 r2β2+ s 2 β2 . (4.9)
Assuming ℜβ2 > 0 i.e. ℜc < 13, this goes to infinity as r, s → ∞. So the infinite sum over
SX,Y converges, and the finite sums that appear in minimal models tend to this infinite
sum in our limit. If however the finite sum is over diagonal states, then the situation is more subtle, because the total dimensions 2∆(r,s) of diagonal degenerate states do not
tend to infinity as r, s → ∞. Our heuristic analysis of the limit of the spectrum may therefore not capture the behaviour of correlation functions, and the limiting spectrum need not necessarily be continuous or even diagonal. Our guess is that such subtleties are absent in four-point functions of diagonal fields, whether these four fields belong to A-series minimal models, or to the diagonal sectors of D-series minimal models. We
expect that these subtleties occur when structure constants are not analytic as functions of momentums. This can happen with the three-point structure constant (3.39) due to its sign factor f2,3(P1), which can be non-trivial if two fields are non-diagonal.
Therefore, we only keep the non-diagonal spectrum S2Z,Z+1
2 as a robust prediction
from minimal models. Actually, a numerical bootstrap analysis in the context of the Potts model has shown that this spectrum appears in correlation functions of the type [9]
Z0 = VPD (0, 12 )V N (0,12)V D P(0, 1 2 ) V(0,N1 2) . (4.10)
Our present analysis suggests that the same spectrum should appear in many more cor-relation functions. These corcor-relation functions should conserve diagonality, as an inher-itance from D-series minimal models. Actually, diagonality must be conserved in any theory whose non-diagonal sector is S2Z,Z+1
2, as a consequence of our condition (3.21) on
three-point functions.
Let us point out that the shift equations (3.13) completely determine the dependence of structure constants on fields in S2Z,Z+12. This is obvious for the dependence on the
first index r, which takes values in 2Z while the relevant equation shifts it by 2. This is less obvious for the dependence on s, because shifts by 2 relate all values s ∈ Z + 1 2
to two values, say s ∈ {−1 2,
1
2}, rather than just one value. However, our two values of
s are opposite to one another, and we can use the fact that normalization-independent quantities are invariant under (r, s) → (−r, −s), because VN
(r,s) and V(−r,−s)N have the same
conformal dimensions. Therefore, the solution of the shift equations is unique up to an (r, s)-independent factor. In order to compute structure constants, we can use indifferently the shift equations, or their solution.
4.2 Numerical tests of crossing symmetry
We conjecture that for any central charge c such that ℜc < 13, for any two diagonal fields with arbitrary momentums, and any two non-diagonal fields in S2Z,Z+1
2, there is a
crossing-symmetric four-point function
Z =DVP1DV(r2,s2N )VP3DV(r4,s4N )E , (4.11) with the spectrum S2Z,Z+1
2 in the s- and t-channels. (However we know neither the
u-channel spectrums of such four-point functions, nor the spectrums of four-point functions of the type VNVNVNVN .)
We will provide evidence for this conjecture by directly testing the crossing symmetry equation (3.7) for four-point functions of the type
Z =DVP1DV(r2,s2N )VP1DV(r2,s2N )E . (4.12) In this case the structure constants D(r,s) are the same in both s- and t-channels, and are
given by eq. (3.58). We perform the renormalization D(r,s)→
D(r,s) D(0, 1 2 ) , in order to have D(0,1 2)= 1 . (4.13)
The Jupyter notebooks used to perform these computations are available on GitHub. Before testing crossing symmetry of generic four-point functions let us verify that our analytic four-point structure constants (3.58) agree with the numerically determined structure constants in [9], for the four-point function Z0 (4.10) in the case c = 0 that
is relevant for critical percolation. The following table shows the values of the first 9 four-point structure constants calculated with each method. The coefficient of variation c(r,s) is an estimate of the precision of the numerical bootstrap determination of the
corresponding structure constant. The last column shows the relative difference between both calculations.
Numerical bootstrap Analytic bootstrap Relative (r, s) D(r,s) c(r,s) D(r,s) differences 0,1 2 1 0 1 0 2,12 0.0385548051 2.4 × 10−9 0.0385548051 8.7 × 10−10 0,32 −0.0212806511 7.6 × 10−9 −0.0212806510 2.9 × 10−9 2,32 0.0004525024 2.2 × 10−8 0.0004525024 1.7 × 10−9 0,52 −3.5638 · 10−5 4.4 × 10−7 −3.5638 · 10−5 3.7 × 10−8 4,1 2 −2.9746 · 10−6 2.4 × 10−6 −2.9746 · 10−6 1.2 × 10−6 2,5 2 8.4077 · 10−7 1.3 × 10−5 8.4078 · 10−7 6.3 × 10−6 4,32 −4.4131 · 10−8 1.6 × 10−4 −4.4135 · 10−8 8.8 × 10−5 0,72 1.5064 · 10−10 9.3 × 10−3 1.5174 · 10−10 7.3 × 10−3 (4.14)
We note that not only the absolute values, but also the signs of the structure constants agree, a result that could not be deduced from the geometric mean formula (1.1). While the precision of the numerical bootstrap calculations decreases as s-channel conformal di-mensions increase, our analytic results do not have this problem. The estimated numerical uncertainty c(r,s) is comparable to the difference with the analytic results, in agreement
with the idea that the analytic results are indeed exact.
Let us now test crossing symmetry of four-point functions computed from our analytic structure constants. For a number of choices of the parameters c, ∆1 and (r2, s2), we will
display the values four-point function Z (or its real part when it is complex) computed from the s- and t-channels for four values of the cross-ratio z, and the relative difference 2 Z(s)−Z(s) Z(s)+Z(t)
between the two channels. Our first choice of parameters corresponds again to the four-point function Z0 at c = 0:
c = 0 ∆1 = ∆(0,1 2) (r2, s2) = (0,12) → z Z0(z) Difference 0.01 s : 0.420743288653023 t : 0.420743500577090 5 × 10 −7 0.03 s : 0.514703102283562 t : 0.514703104911165 5.1 × 10 −9 0.1 s : 0.635102793381169 t : 0.635102793359283 3.4 × 10 −11 0.2 s : 0.706457914575874 t : 0.706457914575509 5.2 × 10 −13 0.4 s : 0.761209621824938 t : 0.761209621824937 8.8 × 10 −16
c = 0.7513 ∆1 = 0.0731 (r2, s2) = (0,32) → z Z(z) Difference 0.01 s : 0.458407080230741 t : 0.458407080231849 2.4 × 10 −12 0.03 s : 0.453858548437590 t : 0.453858548437598 1.8 × 10 −14 0.1 s : 0.166547888308668 t : 0.166547888308669 8 × 10 −15 0.2 s : −0.350128460299570 t : −0.350128460299570 1.4 × 10 −15 0.4 s : −1.085635947272218 t : −1.085635947272219 8.2 × 10 −16 (4.15) c = 0.7513 ∆1 = 0.0731 (r2, s2) = (2,12) → z Z(z) Difference 0.01 s : 0.454365494340930 t : 0.454365494931425 1.3 × 10 −9 0.03 s : 0.488568249965185 t : 0.488568249967384 4.5 × 10 −12 0.1 s : 0.603030367288685 t : 0.603030367288691 9.9 × 10 −15 0.2 s : 0.968890687817652 t : 0.968890687817658 6.6 × 10 −15 0.4 s : 1.720792857730145 t : 1.720792857730149 1.9 × 10 −15 (4.16) c = 4.72 + 0.12i ∆1 = 0.231 + 0.1432i (r2, s2) = (0,12) → z ℜZ(z) Difference 0.01 s : 0.616734633551431 t : 0.616734633551427 1.3 × 10 −14 0.03 s : 0.712923810169360 t : 0.712923810169357 9.9 × 10 −15 0.1 s : 0.784039104794377 t : 0.784039104794372 6.5 × 10 −15 0.2 s : 0.772087133344852 t : 0.772087133344848 5.3 × 10 −15 0.4 s : 0.724037027055028 t : 0.724037027055025 3.9 × 10 −15 (4.17)
c = 4.72 + 0.12i ∆1 = 0.231 + 0.1432i (r2, s2) = (2,32) → z ℜZ(z) Difference 0.01 s : 0.642349222078378 t : 0.642607584797052 4.0 × 10 −4 0.03 s : −0.111756545727778 t : −0.111755334852274 5.5 × 10 −6 0.1 s : −2.187839195998956 t : −2.187839195940312 5.4 × 10 −11 0.2 s : −4.295291665306662 t : −4.295291665306607 1.9 × 10 −14 0.4 s : −13.166871727284267 t : −13.166871727284297 7.6 × 10 −15 (4.18)
We obtain larger differences for some values of the parameters, because of numerical truncations in calculations of conformal blocks. Of course, differences increase as z → 0 where the t-channel expansion diverges. Moreover, values of c ∈ (−∞, 1) suffer from the proximity of poles at the minimal model values c = cp,q. And larger values of ∆1, ∆(r2,s2)
lead to larger errors.
Overall, we find strong evidence that crossing symmetry is satisfied. This supports our claim that there exist consistent rational conformal field theories whose non-diagonal spectrum is S2Z,Z+1
2, and whose structure constants satisfy the analytic bootstrap
equations of Section 3.
4.3 Case of the Ashkin–Teller model
The Ashkin–Teller model provides an example of a four-point function that is known analytically, and does not conserve diagonality. The model has the central charge c = 1, and it has a four-point function of the type [5]
VPD(0, 1 2 ) VPD(0, 1 2 ) V(0,N1 2)V N (0,1 2) = X r∈2Z X s∈Z D(r,s)F∆(s)(r,s) ¯ F∆(s)(r,−s) , (4.19)
with the four-point structure constants D(r,s)= (−1) r 216− 1 2r2− 1 2s2 . (4.20)
Let us see whether this obeys our shift equations (3.13). We first compute the relevant ratios ρ. In the case c = 1, eq. (3.30) reduces to
ρ(V ) = sin(2πσ ¯P )
sin(2πP ) ⇒ ρ(V
N
(r,s)) = (−1)2s . (4.21)
Then let us evaluate eq. (3.25) with β = 1 and P2 = ¯P2 = P3 = ¯P3 = 14. Using the
Gamma function’s duplication formula, and more specifically its consequence Y ±± Γ(12 ± 14 ±14 + x) = 22−4xπΓ(2x)Γ(2x + 1) , (4.22) we find ρ(V1|V2, V3) = (−1)2s22−4(P1+σ1 ¯ P1)sin(2πσ1P¯1) sin(2πP1) . (4.23)
We are interested in two cases of this ratio, where the fields V2, V3 are either diagonal
(thus s2 = 0) or non-diagonal (with s2 = 12). In these two cases, we find
ρV(r,s)N V N (0,12), V N (0,12) = −(−1)2s16−r , ρV(r,s)N V D P(0, 1 2 ) , VPD (0, 12 ) = (−1)2s16−r . (4.24) For non-diagonal fields with s ∈ Z, the shift equation therefore reduces to
D(r+1,s)
D(r−1,s) = −16
−2r , (4.25)
and this is indeed obeyed by the four-point structure constant (4.20). By a similar analysis, we would find the second shift equation D(r,s+1)
D(r,s−1) = 16
−2s, where we no longer have a minus
sign because the non-diagonal field VN
(0,12) has an integer first index.
Our shift equations relate all the structure constants to D(0,0) and D(0,1), but do
not relate these two numbers with one another. Nevertheless, their compatibility with the known structure constants is a non-trivial test of our ideas and calculations. And this case illustrates the difference between diagonal and non-diagonal fields with identical momentums.
5
Conclusion
Our results suggest that for any central charge c such that ℜc < 13, there exists a non-rational conformal field theory, whose spectrum has the non-diagonal sector S2Z,Z+1
2.
The three-point structure constants in that theory are given by our analytic formulas. For c < 1, that theory is a limit of D-series minimal models. Some particular four-point functions in that theory may describe connectivities of clusters in the critical Potts model, in which case our formulas would provide analytic continuations of such connectivities to arbitrary complex values of the Potts model’s number of states.
The three-point structure constants that we have determined should be valid not only in that theory, but actually in any CFT that obeys our assumptions, starting with the existence of two independent degenerate fields. A second theory of the same type can be obtained by β → β1, and its non-diagonal sector is SZ+12,2Z. But not all interesting CFTs
obey our assumptions. For example, the sigma models of [10] have non-diagonal fields whose indices r take fractional values, rather than our half-integer values. This makes it implausible that the degenerate field Vh2,1i exists, as fusion with this field would shift r by
integers. One might still be tempted to use the formula (1.1) for the structure constants, but this would leave us with an undetermined sign and would most probably be wrong. In such a case, a more promising approach would be to renounce analytic formulas, and determine structure constants using the numerical bootstrap method of [9].
Acknowledgments
We are grateful to Benoît Estienne, Yacine Ikhlef and Raoul Santachiara for stimulating discussions and comments on this text.
A
Universality of the geometric mean relation
Here we show that under rather general conditions, there must be a geometric mean relation of the type of (1.1) between structure constants of diagonal and non-diagonal conformal field theories.
A.1 Mathematical statement
Let D+ and D− be two meromorphic differential operators of order n on the Riemann
sphere. Let a non-diagonal solution of (D+, D−) be a single-valued function f such that
D+f = ¯D−f = 0, where ¯D− is obtained from D− by z → ¯z, ∂
z → ∂z¯. Let a diagonal
solution of D+ be a single-valued function f such that D+f = ¯D+f = 0.
We assume that D+ and D− have singularities at two points 0 and 1. Let (Fǫ i) and
(Gǫ
i) be bases of solutions of Dǫf = 0 that diagonalize the monodromies around 0 and 1
respectively. In the case of (Fi+) this means
D+Fi+= 0 , Fi+ e2πiz = λiFi+(z) . (A.1)
We further assume that our bases are such that
∀ǫ, ¯ǫ ∈ {+, −} , F
ǫ
i(z)Fj¯ǫ(¯z) has trivial monodromy around z = 0 ⇔ i = j ,
Gǫ
i(z)Gj¯ǫ(¯z) has trivial monodromy around z = 1 ⇔ i = j .
(A.2)
For ǫ 6= ¯ǫ this is a rather strong assumption, which implies that the operators D+ and D−
are closely related to one another. This assumption implies that a non-diagonal solution f0 has expressions of the form
f0(z, ¯z) = n X i=1 c0iFi+(z)Fi−(¯z) = n X i=1 d0iGi+(z)Gi−(¯z) , (A.3)
for some structure constants (c0
i) and (d0i). Similarly, a diagonal solution fǫ of Dǫ has
expressions of the form
fǫ(z, ¯z) = n X i=1 cǫ iFiǫ(z)Fiǫ(¯z) = n X i=1 dǫ iGiǫ(z)Giǫ(¯z) . (A.4)
We now claim that
if D+ and D− have diagonal solutions, and if moreover (D+, D−) has a non-diagonal solution, then the non-non-diagonal structure constants are geometric means of the diagonal structure constants,
(c0i)2 ∝ c+i c−i , (A.5)
The proof of this statement is simple bordering on the trivial. We introduce the size n matrices Mǫ such that
Fiǫ = n
X
j=1
Mi,jǫ Gjǫ . (A.6)
Inserting this change of bases in eq. (A.4), we must have
j 6= k ⇒
n
X
i=1
cǫiMi,jǫ Mi,kǫ = 0 . (A.7)
For a given ǫ, this is a system of n(n−1)2 linear equations for n unknowns cǫ
i. One way to
write the solution is
cǫi ∝ (−1)idet i′6=i j6=1 Miǫ′,1Miǫ′,j = (−1)i Y i′6=i Miǫ′,1 ! det i′6=i j6=1 Miǫ′,j . (A.8)
Similarly, inserting the change of bases in the expression (A.3) of a non-diagonal solution, we find
j 6= k ⇒
n
X
i=1
c0iMi,j+Mi,k− = 0 . (A.9)
We will write two expressions for the solution of this linear equations,
c0i ∝ (−1)idet i′6=i j6=1 Mi−′,1Mi+′,j = (−1)i Y i′6=i Mi−′,1 ! det i′6=i j6=1 Mi+′,j , (A.10) ∝ (−1)idet i′6=i j6=1 Mi+′,1Mi−′,j = (−1)i Y i′6=i Mi+′,1 ! det i′6=i j6=1 Mi−′,j . (A.11) Writing (c0
i)2 as the product of the above two expressions, we obtain eq. (A.5).
The difficult problem is actually to study the conditions on the matrices Mǫ for
diagonal and non-diagonal solutions to exist. It appears that the existence of non-diagonal solutions is in general equivalent to
∀i, j, Mij+ (M+)−1 ji = M − ij (M−)−1 ji , (A.12)
but this equation is not easy to solve.
A.2 Conformal field theory applications
In our case, the differential operators D+ and D− are respectively left- and right-moving
BPZ equations of the order n = 2. The solutions Fǫ
i and Giǫ are respectively s- and
t-channel conformal blocks, and the matrices M+ and M− are respectively the left- and
right-moving fusing matrices. The structure constants c0
i and d0i are respectively s- and
t-channel four-point structure constants for a four-point function of one degenerate field, and three fields that may be non-diagonal.
Then our mathematical statement is the geometric mean relation (3.60) for such degenerate four-point structure constants. But we have seen in Section 3.1 how these degenerate structure constants determine more general three- and four-point structure constants via shift equations. This is why the geometric mean relation holds in four-point functions that do not involve degenerate fields, although such four-four-point functions violate our assumptions: they are combinations of infinitely many conformal blocks, do not obey differential equations, and violate the assumption (A.2) because for example both F∆(s)(r,s)(z) ¯F (s) ∆(r,s)(¯z) and F (s) ∆(r,s)(z) ¯F (s) ∆(r,−s)(¯z) are single-valued if rs ∈ Z.
Our derivation of the geometric mean relation implies that it holds for four-point functions that obey BPZ equations of any order. It can also hold in CFTs with W-algebras, provided our assumptions are obeyed. The assumption (A.2) may be difficult to satisfy for rational central charges, because conformal dimensions of two fields can easily differ by integers. It looks easier to satisfy when the central charge is generic.
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